Kripke’s Argument Against 4-Dimensionalism

In celebration of my one year in the blogosphere I have decided to start a new (randomly published) series of posts highlighting a post from exactly one year ago that did not recieve much attention. Here is the third  installment.


As I sit here and type this in late 2007, in my living room, it seems to me that all of me is in the room that I am in right now. From a common sense perspective I am entirely in this room; we can put this by saying that I am wholly present in this room. The four-dimensionalist denies this common sense claim. They think that I have some parts, temporal parts, which are not in this room right now. For instance, I have a temporal part that is at work yesterday, or is my child-hood self back in Los Angeles. This is a very odd view indeed! 4-dimensionalism is closely related to eternalism, which Jason has recently defended in an audio post over at Philosophical Pontifications. Kripke has a well known argument which is supposed to show that four-dimensionalism is false. It is a very simple argument which claims that the four-dimensionalist cannot capture certain basic facts about the world. In particular it is not able to capture certain facts about whether a certain disk was spinning in its lifetime or not. Let us look at his very simply argument.

Kripke has argued (in unpublished lectures) that the, as he calls it, Holographic Theory (so-called because the temporal slices are presumed to be three-D, like holograms) has serious problems by suggesting that it is unable to model certain facts about the history of the world. His argument relies on two thought experiments. First, consider a disk made from uniform material. It is a single color and is made from entirely continuous matter. Now, a disk modeled in three dimensions (which turns into a cylinder as it ‘progresses’ through the third dimension) is akin to modeling a three dimensional object in time. Any problems that our three-D model of the disk will have will be simplified versions of problems for the view that objects are four-dimensional. So, now what if we are looking at the resulting cylinder, which represents the disk’s ‘life’, and we wonder ‘was the disk spinning during its life?’ 

The temporal stage theory is unable to answer this question. We cannot answer the question because the model of the disk’s ‘life’ is incapable of representing this kind of fact. What we need in order to answer the question is an answer to the question ‘is some specific part of the disk now in another place?’ The same thing being at a different place later is what motion is. Since we have no criterion to determine if we have the same substance in the same place we cannot, in principle, answer the question about the disk’s motion. Presumably the disk was either spinning or not and since there is no way to account for this, one way or the other, the temporal slice theory is in trouble. The second thought experiment is quite similar and involves an infinite river. He introduces this variation only so that he may avoid the objection that a disk made from uniform matter is impossible. The water in the river takes care of this. We need not rehearse it here, as it is basically a reiteration of the previous argument applied to the river.  Kripke concludes that the temporal slice theory is wrong and suggests we that need a primitive notion of the identity of substance.

This really should not come as much of a surprise as it is clear that Kripke had this idea in mind when he delivered the lectures now known as Naming and Necessity. He argues that we can stipulate that we are talking about certain substances and ask what would happen to them in a possible world. This is only possible because there are substances out there with which we have interacted. It is because I have interacted with Kripke that I can ask what would happen to him, Saul Kripke, had he not been interested in philosophy. Similarly I can stipulate that I am talking about this table right here and ask about what happened to it yesterday, or what might happen to it three days from now. The argument that Kripke advanced against 4-d-ism is merely making explicit something that has been implicitly accepted by anyone who finds Kripke’s arguments compelling; Namely that we must take the logical relation of the identity of substance (A=A) as a primitive notion.

To illustrate Kripke talks about possible states of two six-sided die. The fundamental question is what we are to say in the cases where we get a four and a three, say. Is there just one state or are there two? Kripke argues that each possible state of the dice is distinct. These states will be distinct because the two die are enduring substances that we can rigidly designate. So each die will have a unique state that it can be in or not. Let us call the two die Alice and George. Then there is one possible state where Alice is four and George is three and another possible state where George is four and Alice is three. These two states are distinct even though the two dice, and so the two states, are qualitatively identical. Kripke’s basic admonishment is that we cannot count possible states of an object simply by qualitative similarity, as that collapse the number of possibilities.

In the next post I want to address Ted Sider’s temporal counter-part theory as a response to Kripke’s argument.


25 thoughts on “Kripke’s Argument Against 4-Dimensionalism

  1. Hi Richard,

    Thanks for linking. It’s nice to see someone’s reading my blog.

    As for the substance of your post, Craig Callender has an interesting paper questioning the notion of homogenous rotating discs, only his challenge doesn’t depend on rejecting homogenity:

    Also, it can be questioned whether we really can stipulate the distinctness of possible states in the way Kripke thinks we can. This article on identity and individuality in quantum mechanics suggests that for subatomic particles there really is a collapse in the number of possibilites:

    Hope you find these interesting.

  2. Thanks Jason for the comment…I looked at the Callender paper (briefly)…I am not sure that his argument address the concern I am raising here. He is worried about the Humean Superveinence claim, whereas I am worried about 4-dimensionalism…Lewis could be right about this claim and yet 4-d’ism wrong…

    I will, however, look over the paper in more detail and get back to you…same for the quantuum stuff…

  3. Hmm… well, it has been a while since I read Callender’s paper, and I apologize if it doesn’t adress your concern. But I do think the one on identity in quantum theory is directly relevant to Kripke’s idea that one can simply stipulate the distinctness of possible states.

  4. Hey Clark, thanks for the comment, and sorry for the delay in getting back to you…I have been super busy out here in meatspace!

    In order to include such variables we would already have to have an answer tot he wuestion of whether the disk was spinning or not and so to include them is just to beg the question. What we need is someway to determine the momentum from the time slices that we have.

  5. But Richard that avoids the question of what the time slices are of. If one simply says all that exists is matter and the properties of matter (like momentum) come on top that is itself a huge assumption. One could simply say that momentum or other state variables are intrinsic to the matter. So you can’t talk about the matter without talking about the state variables.

    Put an other way it seems that the question itself is sneakily assuming certain conclusions and may in fact be circular or at bare minimum attacking a strawman.

  6. Hey Clark,

    yeah that’s a good point. I went back and looked at my notes from Kripke’s class and I think I found an argument that velocity won’t be included in the slices. A time-slice will by definition include onloy the properties which an object has at that instant, but velocity is the derivitive of the object’s speed and positions over some interval of time…but the future properties of the object have no influence over its present properties, nor do the present properties have any influence over the past properties. So velocity is not the kind of property that will figure in an individual time slice. Since momentum is mass times velocity, neither will momentum.

  7. Richard, that’s a common misunderstanding of velocity. It’s not accurate. It confuses velocity as measured from velocity as a state variable. But even calculus, whatever its philosophical ground, v(t) = dx/dt where dx and dt are infintisimals. This might be a problem except that the thought experiment to work requires the slice to be dt.

  8. To add, in QM, momentum is typically taken as a fundamental state. It gets complex of course so for anyone who gets pedantic I know I’m oversimplifying.

    The point is that your rejoinder just doesn’t work.

  9. It confuses velocity as measured from velocity as a state variable.”

    Uh, no. It’s an argument that there is no such thing as ‘velocity as a state variable’. Velocity is a ‘cross-temporal-slice’ property…unless, that is, you have some argument for your claim?

  10. RB,
    but treating velocity as a state variable makes it like all those other variables and thus simpler (occams razor). anyway if a time slice did not contain a trait that amounted to ‘momentum’ how would the next time slice know what to do?

  11. Richard, I think my point is that if one wants to make a convincing argument one ought deal with what physics treats as state variables. And physics treats velocity, among others, as a state variable. Now one can of course debate whether physics ought so treat them. And then into the associated debate about scientific realism. But those arguments are probably more complex than you want to get into.

    My argument is just that by an appeal to physics there is prima facie reasons to think that Kripke’s argument works only by excluding any physical properties beyond mass in temporal slices. To ask for more argument than that is really to make a burden or proof argument. But I think that while philosophers tend to like to talk about time and space in more Cartesian ways that the burden of proof will always be on those who wish to talk in a fashion different from physics.

  12. To add if one goes into physics there are some problems on the face of it with the very notion of temporal slices. After all in QM one has to ask whether one can make any sense of the notion of a discrete temporal slice. (There are some ways but they tend to be controversial) The simplest form of this problem is uncertainty over time which makes a definite temporal slice problematic.

    Move to GR and of course you then face the problem that a straightforward reading of GR imposes 4-dimensionalism. There are attempts to argue against this although most of the arguments I’ve read limit themselves to Special Relativity or else end up being unpersuasive when dealing with the physics. (Of course what is or isn’t persuasive varies from person to person – and there definitely are anti-eternalist arguments in the context of GR)

    My point isn’t to make an argument about 4D proper just to point out that Kripke’s particular solution seems to make some problematic assumptions about physics.

  13. Hi Richard,

    You say: “…but the future properties of the object have no influence over its present properties, nor do the present properties have any influence over the past properties.”

    While the dependence of present truths or properties on future ones may seem strange, I think there are examples for which it makes intuitive sense. Suppose that someone writes down the following statement at some time prior to 12:00pm on the day it is written down:

    (1) “Zeno’s arrow is flying through the air at 12:00pm.”

    If we regard 12:00pm as an instant, a duration of zero extent, then there is no intrinsic difference between the arrow’s moving at 12:00pm and its hovering in the air while being at rest at 12:00pm. (Given what you’ve said about velocity, I’m sure you’ll agree with this.) Since there is no intrinsic difference, however, it seems to me that (1)’s truth or falsity depends on whether or not Zeno’s arrow *will* successively occupy different points in space in the very near future. If we accept eternalism, there is no problem, for the future is “already there.” But if its truth depends on what will happen in the future, and we *deny* eternalism, then at 12:00pm (1) is neither true nor false (because there is as yet no future to make it true or false), although it would appear that once 12:00pm has passed it retroactively *becomes* true that Zeno’s arrow *was* flying through the air (or not) at that time (because after 12:00pm Zeno’s arrow either will or will not have successively occupied different points in space). I can’t speak for others, but for me that result seems even stranger than the admittedly strange dependence of present truths or properties on future ones.

  14. Oops, above I should have said:

    “Since there is no intrinsic difference, however, it seems to me that (1)’s truth or falsity *at 12:00pm* depends on whether or not Zeno’s arrow *will* successively occupy different points in space in the very near future.”

  15. Hi Clark, sorry for the long delay in getting back to you.

    I think you raise an interesting point, but I am not convinced. Physics often employs simplifying assumptions (like that mass is an intrinsic property of an object in Newtonian mechanics) but that isn’t any reason to think that that is the way that things are. Why can’t I make the same response to your point about velocity being treated as a state variable? It is useful to do so, in the context of some physical theory, but why think it is any more than that, especially given the above argument?

  16. Hi Jason, very nice point.

    First I wonder if you are begging the question by assuming that 12 p.m. is a durationless instant when you are denying eternalism. I take it that if eternalism is false then time won’t be discrete in this way, though I may be wrong…

    Second, I just don’t share your sense of strangeness here. If the future is unwriten (sorry, I have been doing a lot of Terminator stuff lately) then what could possibly account for (1)’s truth (or falisty)? It seems to me that you will find this strange only if you are already entreanched in an eternalist viewpoint.

  17. BTW Richard, someone over in my thread posted an interesting link arguing against instantaneous velocity in classical physics. It took me a while to be able to download the article from JSTOR. It’s quite good although I’m far from convinced. However he doesn’t address quantum theory where momentum has absolute status if it is the eigenvalue of an eigenvector. On the other hand someone probably is out there arguing against that position as well. But that is orthodox quantum theory.

  18. Richard,

    Regarding your first response, I find it hard to see how time could *fail* be discrete in that way, even on an A-theoretic understanding. If, for example, the future is “unwritten”, then the present cannot be temporally extended, for if the present were extended it would be an interval which is composed of an earlier and a later portion. In other words, some portion of the present would lie in the future of another portion, and if that were so some portion of the future would be “written” after all. And if the present isn’t temporally extended, isn’t it a durationless instant?

    As for your second response, let me try and bring out the sense of strangeness this way: Suppose you are right about the future’s being unwritten. When could (1) be true? Certainly not prior to 12:00pm on that day. Nor could it be true *at* 12:00pm on that day, because (1) depends for its truth on what Zeno’s arrow *will* do in the immediate future, and that is unwritten. Could it be true after 12:00 pm? That’s tricky. Once 12:00pm has passed it seems to be true that Zeno’s arrow either was or was not flying through the air at 12:00pm, but then we have a clash with what was just said, for how can it be true that Zeno’s arrow *was* flying through the air (or not) at 12:00pm if the statement that Zeno’s arrow *is* flying through the air at 12:00pm is/was without truth value at 12:00pm?

  19. Hey Jason,

    RE 1: I don’t see the problem there. It could be that the present is extended, and the ‘later’ point –the future– is caused by the present. So, it is written, but only a bit at a time and ‘on the fly’

    RE 2: The claim is either true or flase at 12:00, though we cannot determine which it is until just after 12:00. What’s wrong with this answer?

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